WISSENSCHAFTSZENTRUM BERLIN FÜR SOZIALFORSCHUNG
SOCIAL SCIENCE RESEARCH CENTER BERLIN
ISSN Nr. 0722 – 6748
Research Area
Markets and Political Economy Research Unit
Market Processes and Governance
Forschungsschwerpunkt Markt und politische Ökonomie Abteilung
Marktprozesse und Steuerung
Kai A. Konrad * Dan Kovenock **
Equilibrium and Efficiency in the Tug-of-War
* WZB and Free University of Berlin
** Purdue University
SP II 2005 – 14
August 2005
Zitierweise/Citation:
Kai A. Konrad, Dan Kovenock,Equilibrium and Efficiency in the Tug-of-War, Discussion Paper SP II 2005 – 14, Wissenschaftszentrum Berlin, 2005.
Wissenschaftszentrum Berlin für Sozialforschung gGmbH,
Reichpietschufer 50, 10785 Berlin, Germany, Tel. (030) 2 54 91 – 0 Internet: www.wz-berlin.de
ABSTRACT
Equilibrium and Efficiency in the Tug-of-War by Kai A. Konrad and Dan Kovenock *
We characterize the unique Markov perfect equilibrium of a tug-of-war without exogenous noise, in which players have the opportunity to engage in a sequence of battles in an attempt to win the war. Each battle is an all-pay auction in which the player expending the greater resources wins. In equilibrium, contest effort concentrates on at most two adjacent states of the game, the "tipping states", which are determined by the contestants' relative strengths, their distances to final victory, and the discount factor. In these states battle outcomes are stochastic due to endogenous randomization. Both relative strength and closeness to victory increase the probability of winning the battle at hand. Patience reduces the role of distance in determining outcomes.
Applications range from politics, economics and sports, to biology, where the equilibrium behavior finds empirical support: many species have developed mechanisms such as hierarchies or other organizational structures by which the allocation of prizes are governed by possibly repeated conflict. Our results contribute to an explanation why. Compared to a single stage conflict, such structures can reduce the overall resources that are dissipated among the group of players.
Keywords: Winner-take-all, all-pay auction, tipping, multi-stage contest, dynamic game, preemption, conflict, dominance
JEL Classification: D72, D74
* Comments by Daniel Krähmer and Johannes Münster are gratefully acknowledged. Part of this work was completed while the second author was Visiting Professor at the Social Science Research Center Berlin (WZB). The usual caveat applies.
ZUSAMMENFASSUNG
Gleichgewicht und Effizienz im "Tug of War"
Wir beschreiben das eindeutige Markoff-perfekte Gleichgewicht in einem mehrstufigen Konflikt ohne exogene Unsicherheit ("noise"), bei dem die Spieler versuchen, in einer Serie von aufeinander folgenden kleineren Gefechten einen Konflikt zu gewinnen. Jedes Gefecht ist eine "all-pay auction", bei der derjenige Spieler gewinnt, der die meisten Ressourcen eingesetzt hat. Im Gleichgewicht konzentriert sich der Mitteleinsatz auf höchstens zwei benachbarte Zustände, die wir als spielentscheidende Zustände ("tipping states") bezeichnen. Die Lage dieser Zustände hängt von der relativen Stärke der Spieler, der Zahl der ver- bleibenden Spielstufen bis zum Gesamtsieg und dem Diskontierungsfaktor ab.
An diesen kritischen Zuständen ist der Konfliktausgang zufällig aufgrund der stochastischen Verteilung der im Gleichgewicht gewählten Mengen von Konflikt- ressourcen. Sowohl die relative Stärke als auch die Nähe zur finalen Kon- fliktstufe erhöhen die Wahrscheinlichkeit, das einzelne Gefecht zu gewinnen.
Geringe Kosten des Wartens verringern den Einfluss der Entfernung zum Gesamtsieg auf den Ausgang der einzelnen Gefechte. Die Anwendungsgebiete sind zahlreich und reichen von der Politik über die Wirtschaft und den Sport bis zur Biologie. Dort findet das Gleichgewichtsergebnis empirisch Unterstützung:
Viele Arten haben eigene Mechanismen entwickelt, z.B. Hierarchien oder andere Organisationsstrukturen, bei denen die Allokation der Siegerprämie in sich möglicherweise wiederholenden Konflikten erfolgt. Unsere Ergebnisse liefern hierzu eine Erklärung. Im Vergleich mit einem einstufigen Konflikt können solche Strukturen den Ressourceneinsatz der Spieler reduzieren.
1 Introduction
Final success or failure in a conflict is often the result of the outcomes of a series of potential battles. An illustrative example is the decision making process in many organizations. Resources, jobs and other goods that invol- ve rents to individuals inside the organization are frequently allocated in a process that has multiple decision stages. For instance, hiring decisions often involve a contest between candidates in which a hiring committee makes a decision and forwards this decision to another committee. This committee approves to the initial decision and forwards the case further until a final decision stage is reached, or may return the case to the previous committee.
Candidates could expend effort trying to influence the decision process in each stage, but if at all, typically serious efforts are expended by the candi- dates only in early stages of the decision process. Such multi-layered decision processes obviously cause delay in decision making and this can be seen as a cost. We will argue here that, compared to a single stage decision process in which the rival players spend effort in a single stage all-pay auction, the multi-stage decision process can be advantageous as it may improve alloca- tive efficiency and reduce effort that is expended by rival contestants in the conflict.
In more general terms we describe the multi-stage contest as a tug-of- war. As a modeling device, the tug-of-war has a large number of applications in diverse areas of science, including political science, economics, astronomy, history and biology.1 It consists of a (possibly infinite) sequence of battles between two contestants who accumulate stage victories, and in which the
1To give a few examples: In politics, Whitford (2005) describes the struggle between the president and legislature about the control of agencies as a tug-of-war. Yoo (2001) refers to the relations between the US and North-Korea and Organski and Lust-Okar (1997) to the struggle about the status of Jerusalem as cases of tug-of-war. According to Runciman (1987), at the time of the Crusades, when various local rulers frequently attacked one another, they sometimes succeeded in conquering a city or a fortification, only to lose this, or another, part of their territory to the same, or another, rival ruler later on. The conflict between two rival rulers can be seen as a sequence of battles. They start at some status quo in which each rules over a number of territories with fortified areas. Theyfight each other in battles, and each battle is concerned with one fortress or territory. In the sequence of successes and failures, the fortresses or territories are destroyed or reallocated, and the conflict continues until one of the rulers has lost all his fortresses or territories and is thus finally defeated. If battle success alternates more or less evenly, then such a contest can go on for a very long time, possibly even forever. The end comes only when one of the rulers has been more successful than his rival sufficiently often.
contestant whofirst accumulates a sufficiently larger number of such victories than his rival is awarded the prize for final victory.2
To our knowledge, Harris and Vickers (1987) were the first to look for- mally at the tug-of-war. They analyse an R&D race as a tug-of-war in which each single battle is determined as the outcome of a contest with noise. Such exogenous noise makes the problem less tractable and has so far ruled out a fully analytic description of the equilibrium. Budd, Harris and Vickers (1993) apply a somewhat more complicated stochastic differential game approach to a dynamic duopoly, seen as a tug-of-war involving a continuum of advertizing or R&D battles that determine the firms’relative market positions. Using a complementary pair of asymptotic expansions for extreme parameter values and numerical simulations elsewhere, they isolate a number of effects that govern the process. Several of these appear in our analysis which, unlike their framework, derives an analytical solution for the unique Markov per- fect equilibrium. Morever, our analysis explicitly solves for equilibrium for both symmetric and asymmetric environments.
The term ‘tug-of-war’ has also been used in biology. In the context of within-group conflict among animals, subjects could struggle repeatedly.3 For instance, the formation of hierarchies and their dynamic evolution occurs in repeated battle contests. As Hemelrijk (2000) describes for several examp- les, individuals may try to acquire a high rank, but the differentiation and asymmetry that is created by this can also reduce future conflict. Winning or losing a particular contest in a series of conflictual situations is known to change future conflict behavior (Bergman et al. (2003), Beacham (2003) and Hsu and Wolf (1999)). This may partially be the result of information about own fighting skills and the fighting experience gained, but it may also arise
2According to Wikipedia the term tug-of-war refers to a rope pulling contest in which two contestants (or groups) pull a rope in different directions until one of the sides pulled the rope (and the opponent group) across a certain limit. In more abstract terms, the contest consists of a series of battles, where a battle victory of one player makes both move one unit towards the winner’s preferred terminal state, and where one contestant wins the war if the difference between the winner’s number of such battle victories exceeds the other contestant’s number of battle victories by some absolute number.
3The term also refers to contests between different species. Ehrenberg and McGrath (2004) refer to the interaction of microtubule motors, Larsson, Beignon and Bhardwaj (2004) and Zhou et al. (2004) refer to the interaction between viruses and the dendritic cells or other parts of the immune system as tugs of war. Tibbetts and Reeve (2000) consider the role of the amount of reproductive sharing within a group for the likelihood of within-group conflict among the social wasp Polistes dominulus.
from the change in strategic position with respect to future conflict about rank, territory, access to food, or opportunities to reproduce.
Evidence from biology and political science shows that violent conflict often does not take place, or, at least, the intensity of a conflict varies si- gnificantly as a function of the conflicting parties’ actual strengths, previous experience, and the strategic symmetry or asymmetry of the particular si- tuation in terms of territorial or other advantages. Parker and Rubenstein (1981) and Hammerstein (1981) emphasize the role of asymmetry in determi- ning whether a conflictual situation turns into a resource wasteful or violent conflict. Different advantages and disadvantages may determine the over- all asymmetry of a conflictual situation, and counterbalance or add to each other. Schaub (1995) describes the conflict over food that occurs between long-tailed macaque females. Differences in strength and in the distances between the animals and the location of the food govern their behavior. Su- perior strength or dominance of one contestant can be compensated by a greater distance she has to the location of the food. Relative strength, to- gether with the actual payoffs from winning determine contestants’ stakes at any given stage of a tug-of-war and determine the degree of asymmetry between the rival players.
We examine how the players’ respective fighting abilities, rewards from final victory, and the distances in terms of the required battle win differential to achieve victory interact to determine Markov perfect equilibrium behavior in the tug-of-war. For notational convenience we concentrate on the asymme- try in the valuations of the final prize and assume equal fighting ability, but as will be shown this is equivalent to the more general case with asymmetric valuations of the prize and asymmetric fighting abilities. We show that the contest effort that is dissipated in total and over all battle periods crucial- ly depends on the starting point of the tug-of-war, and, for many starting points, is negligible, even if the asymmetry in the starting conditions is very limited. Hence, the multi-battle structure in a tug-of-war reduces the amount of resources that is dissipated in the contest, compared to a single all-pay auction, which has been studied by Hillman and Riley (1989) and Baye, Ko- venock and deVries (1993, 1996) for the case of complete information and by Amann and Leininger (1995, 1996), Krishna and Morgan (1997), Kura (1999), Moldovanu and Sela (2001) and Gavious, Moldovanu and Sela (2002) in the context of incomplete information.4
4For further applications of the all-pay auction see Arbatskaya (2003), Baik, Kim and
Our results may contribute to explaining why mechanisms such as hierar- chies or other organizational structures have evolved by which the allocation of prizes is governed by a multi-stage conflict.5 Such structures may delay the allocation of a given prize, compared to a single stage conflict, but can con- siderably reduce the overall resources that are dissipated among the group of players. Compared to a standard all-pay auction, a tug-of-war that is not rigged in favor of one of the players also improves allocative efficiency; the probability with which the prize is awarded to the player who values it more highly is higher in the tug-of-war than in the standard all-pay auction.
In the next section we outline the structure of the tug-of-war and cha- racterize the unique Markov perfect equilibrium. In section 3 we discuss the efficiency properties of the tug-of-war and compare it with the all-pay aucti- on. Section 4 concludes.
2 The analytics of the tug-of-war
A tug-of-war is a multi-stage game with a potentially infinite horizon which is characterized by the following elements. The set of players is {A, B}. The set of states of the war is given by a finite ordered grid of m + 1 points M ≡ {0,1, ...m} in R1. The tug-of-war begins at time t = 1 with players in the intitial state j(1) = mA, 0 < mA < m, which may either be chosen by nature, or may be a feature of the institutional design. In each period t = 1,2,3... a battle takes place between the players in which A (resp. B) expends effort at (resp. bt). A victory by player A (B) in state i at time t moves the war to state i−1 (i+ 1) at timet+ 1. The state in periodt+ 1is therefore j(t+ 1) =mA+nBt−nAt, where nAt andnBt denote respectively,
Na (2001), Baye, Kovenock and De Vries (2005), Che and Gale (1998, 2003), Ellingsen (1991), Kaplan, Luski and Wettstein (2003), Konrad (2004), Moldovanu and Sela (2004), and Sahuguet and Persico (2005).
5There are, of course, other explanations for hierarchies more generally, which, however, focus on different aspects of a hierarchy (see, e.g., the survey in Radner 1992). Radner (1993) for instance, considers a problem of efficient information aggregation, asking what is the efficient decision tree. Closer to the issue of allocation of goods in a conflict, Wärneryd (1998) and Müller and Wärneryd (2001) consider distributional conflict between rival groups followed by distributional conflict within the winning group as a type of hierarchical conflict. Both these approaches focus on the "tree-ör "pyramidproperty of hierarchies that reduces the number of players when moving to the top, whereas our approach does not use this property. We consider only two contestants throughout and focus on the sequential, repeated nature of decision process.
the number of battle victories that A and B have accumulated by the end of period t. This continues as long as the war stays in some interior state j ∈Mint ≡{1,2, ...,(m−1)}. The war ends when one of the players achieves
final victory by driving the state to his favored terminal state, j = 0 and
j =m, for playerAand B respectively. A prize (forfinal victory) of sizeZA
is awarded to A if the terminal state j = 0 is reached and, alternatively, a prize of ZB is awarded toB if the terminal statej =m is reached. Without loss of generality we assume that ZA≥ZB. Figure 1 depicts the set of states.
states j m
m-1 mA-1 mA mA+1
0 1
... ...
Figure 1:
Player A’s (B’s) period t payoff πA(at, j(t)) (πB(bt, j(t))) is assumed to equal ZA (ZB) if player A (B) is awarded the prize in that period, and
−at (−bt) if t is a period in which effort is expended.6 We assume that each player maximizes the expected discounted sum of his per-period payoffs.
Throughout we assume that 0< δ <1 denotes the common, time invariant, discount factor.7
The assumption that the cost of effort is simply measured by the effort itself is for notational simplicity only. Since a player’s preference over income streams is invariant with respect to a positive affine transformation of utility, if player A (B) has a constant unit cost of effort cA (cB) we may normalize utility by dividing by cA (cB) to obtain a new utility function representing the same preferences in which the unit cost of effort is1but playerA(B)has a prize value ZA/cA (ZB/cB). Therefore, our model with asymmetric prizes can be interpreted as one with both asymmetric prizes andfighting abilities.
Each single battle in the tug-of-war is a simultaneous move all-pay auc- tion with complete information. A player’s action in each period in which the state is interior is his effort, at ∈ [0, K] and bt ∈ [0, K], for A and B,
6Since the per-period payoffs do not depend directly on time, we have dropped a time index.
7It is straightforward to extend our results to cases in which players have different, time invariant discount factorsδA andδB.
respectively, where K ≥ ZA.8 The player who spends the higher effort in a period wins the battle. We choose a deterministic tie-breaking rule for the case in which both players choose the same effort, by which the ”advantaged”
player wins. Given m, ZA, ZB andδ, we say that playerA is advantaged in state j if δjZA> δm−jZB, andB isadvantaged ifδjZA ≤δm−jZB. We define j0 = min{j ∈ Mint¯¯δjZA ≤δm−jZB} where this is non-empty, and j0 = m otherwise: player B is advantaged for j ∈ Mint such that j ≥ j0 and A is advantaged otherwise.
If m = 2 and mA = 1, the tug-of-war reduces to the well-known case of the standard all-pay auction with complete information at time t = 1, as in Hillman and Riley (1989), Ellingsen (1991) or Baye, Kovenock and deVries (1996). In this case, one single battle takes place at state j =mA = 1. The process moves from this state in period 1 toj = 0or toj = 2at the beginning of period 2, and the prize is handed over toAorB, respectively. Accordingly, the contest at period t = 1 in state j = 1 is over a prize that has a present value ofδZAandδZBforAandB, respectively, and the payoffs in the unique equilibrium of this game (which are in nondegenerate mixed strategies) are δ(ZA−ZB)for Aandzerofor B. In what follows, we consider the case with m >2.
For each period t, if a terminal state has not yet been reached by the beginning of the period, players simultaneously choose efforts with common knowledge of the initial statemAand the full history of effort choices, denoted as (at−1,bt−1) ≡ ((a1, ..., at−1), (b1, ..., bt−1)). Players also know the current state j(t)of the war and the state in any past period j(τ), τ < t. We define jt = (j(1), j(2), ...j(t)), where j(1) = mA. Hence, we will summarize the history at time t along any path which has not yet hit a terminal state by ht = (at−1,bt−1,jt). We will call such a path a non-terminal period t history and will denote the set of such histories by Ht. A history of the game that generates a path that reaches a terminal state at precisely periodtis termed a terminal period t history. Denote the set of terminal period t histories by Tt, and the set of (at−1,bt−1)generating elements of Tt byTet.
If for an infinite sequence of effort choices, a = (a1, a2, ...) and b = (b1, b2, ...) no terminal state is reached in finite time, we will call the cor- responding history h∞ = (a,b,j) anon-terminal history and denote the set
8This upper limit makes the set of possible effort choices compact, but does not lead to a restriction that could be binding in any equilibrium, as an effort choice larger than ZA in some period is strictly dominated by a choice of effort of zero in this and all future periods.
of such histories as H∞.
Given these constructions, we define a behavior strategy σl for player l ∈{A, B} as a sequence of mappings σl(ht) :Ht→Σ[0,K], that specifies for every period t and non-terminal history ht an element of the set of probabi- lity distributions over the feasible effort levels[0, K]. Each behavior strategy profile σ = (σA, σB) generates for eacht a probability distribution over his- tories in the set S
τ≤tTτ ∪Ht. It also generates a probability distribution over the set of all feasible paths of the game, S∞
τ=1Tτ ∪H∞.
Since we assume that each player’s payofffor the tug-of-war is the expec- ted discounted sum of his per-period payoffs, the payofffor player A from a behavioral strategy profile σ is denotedvA(σ) = Eσ(Σ˜tt=1δt−1πA(at, j(t)))≡ Eσ(πA(a˜t−1,b˜t−1,j˜t))where˜t is the hitting time at which a terminal state is first reached.9 If a terminal state is never reached, ˜t = ∞. Note that for a given sequence of actions (a˜t−1 b˜t−1),˜t arises deterministically, according to the non-random transition rule embodied in the all-pay auction, so that the randomness of ˜t is generated entirely by the non-degenerate nature of the probability distributions chosen by the behavioral strategies. If ht+1 = (at
bt,jt+1)∈ Tet+1denotes a sequence of efforts that leads to a terminal state at precisely period ˜t=t+ 1, then, the payoffs for A andB are
πA((at,bt,jt+1)) =
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
− Xt
i=1
δi−1ai +δtZA if j(t+ 1) = 0
− Xt
i=1
δi−1ai if j(t+ 1) =m
(1)
and, respectively,
πB((at,bt,jt+1)) =
⎧⎪
⎪⎪
⎪⎨
⎪⎪
⎪⎪
⎩
− Xt
i=1
δi−1ai +δtZB if j(t+ 1) =m
− Xtˆ
i=1
δi−1ai if j(t+ 1) = 0.
(2)
9We adopt a notational convention throughout this paper that the action set available to each player in a terminal state is the effort level zero, so that for any hitting time ˜t, a˜t =b˜t= 0. Hence, in these statesπA(at, j)andπB(bt, j)include only the prize awarded to the victor, and we suppress the termsat˜andb˜tin the notationΣ˜tt=1δt−1πA(at, j(t))≡ πA(a˜t−1,b˜t−1,jt˜).
If for an infinite sequence of effort choices,a= (a1, a2, ...)andb= (b1, b2, ...) no terminal state is reached in finite time, payoffs are
πA((a,b,j)) = − X∞
t=1
δt−1at and πB((a,b,j)) =− X∞
t=1
δt−1bt.
For a given behavior strategy profileσ= (σA, σB)each player’s payoffin the tug-of-war can be derived from calculating the expected sum of discounted per period payoffs generated by the probability distribution over histories in the set S∞
τ=1Tτ∪H∞. Moreover, for anyt andht∈Ht, one may define each player’s expected discounted value of future per-period payoffs (discounted back to time t) conditional on the history ht by deriving the conditional distribution induced byσ|ht over S∞
τ=t+1Tτ ∪H∞. We shall refer to this as a player’s continuation value conditional on ht and denote it by vi(σ|ht) = Eσ|ht(Σ˜ts=tδs−tπA(as, j(s))). Note that this has netted out any expenditures accrued on the history ht.
Since the players’ objective functions are additively separable in the per- period (time invariant) payoffs and transitions probabilities depend only upon the current state and actions, continuation payoffs from any sequence of current and future action profiles depend on past histories only through the current state j. It therefore seems natural to restrict attention to Markov strategies that depend only on the current state j and examine the set of Markov perfect equilibria. Indeed, this partition of histories is that obtained from the more formal analysis of the determination of the Markov partition in Maskin and Tirole (2001). For anyt, we may partition past (non-terminal) histories inHtby the periodtstatej(t), inducing a partitionHt(·), and define the collection of partitions,H(·)≡{Ht(·)}∞t=1. It can be demonstrated that in our game the vector of collections(HA(·), HB(·)) = (H(·), H(·))is the unique maximally coarse consistent collection (the Markov collection of partitions) in the sense of Maskin and Tirole (2001, p. 201). For any timet,the current state j(t) therefore constitutes what they call the payoff-relevant history. Since our game is stationary, we may partition the set of all finite non-terminal histories by the same state variables, j ∈Mint≡{1,2, ...(m−1)}, removing any dependence of the partition on the timet. We label this partition{j(t) = i}i∈Mint. This is the stationary partitiondefined by Maskin-Tirole, (2001, p.
203).
In the continuation, we restrict attention to (stationary) Markov strate- gies measurable with respect to the payoff relevant history determined by
the stationary partition {j(t) = i}i∈Mint. A stationary Markov strategy σl
for player l ∈ {A, B} is a mapping σl(j) : Mint → Σ[0,K], that specifies for every interior state j a probability distribution over the set of feasible ef- fort levels [0, K]. If in the continuation game starting in period t and state j, σ = (σA, σB) is played, then the continuation value for player i at t is denoted as vi(σ|j) and can be calculated as the discounted sum of future expected period payoffs in a well-defined manner similar to that described above.
In this context we are interested in deriving the set of Markov perfect equilibria; that is a pair of Markov strategies that constitute mutually best responses for all feasible histories. In Propositions 1-3 below we demonstrate that the tug-of-war has a unique Markov perfect equilibrium for any combi- nation of mA, m, ZA, ZB and δ.
Before stating these propositions, it is useful to derive some simple proper- ties that must hold in any Markov perfect equilibrium of our model. Suppose σ∗ = (σ∗A, σ∗B)is a Markov perfect equilibrium and denote playeri’s continua- tion value in state j under σ∗ by vi(σ∗|j) =vi(j). Subgame perfection and stationarity imply that competition in any state j, j ∈ {1,2, ...m−1}, may be viewed as an all-pay auction with prizezA(j) =δvA(j−1)−δvA(j+ 1)for player A and zB(j) = δvB(j + 1)−δvB(j−1)for player B. In equilibrium, the continuation value to player l of being in statej at time t is equal to the sum of the value of conceeding the prize without afight (and thereby moving one state away from the player’s desired terminal state) and the value of engaging in an all-pay auction with prizes zA(j) = δvA(j −1)−δvA(j+ 1) for player A and zB(j) = δvB(j + 1)−δvB(j −1) for player B. An imme- diate consequence of the characterization of the unique equilibrium in the two-player all-pay auction with complete information (see Hillman and Riley (1989) and Baye, Kovenock, and De Vries (1996)) is that local stategies are uniquely determined and the continuation value for the two players in any state j ∈{1, ...m−1} at any timet is
vA(j) =δvA(j+ 1) + max(0, zA(j)−zB(j)) =δvA(j+ 1)+
+ max(0, δ[(vA(j−1)−vA(j+ 1))−(vB(j + 1)−vB(j−1))]) (3) and
vB(j) =δvB(j−1) + max(0, zB(j)−zA(j)) =δvB(j−1)+
+ max(0, δ[(vB(j+ 1)−vB(j −1))−(vA(j −1)−vA(j+ 1))]). (4)
Rearranging (3) and (4) we obtain
vA(j) =δvA(j+1)+max(0, δ[(vA(j−1)+vB(j−1))−(vA(j+1)+vB(j+1))]) (5) and
vB(j) =δvB(j−1)+max(0, δ[(vA(j+1)+vB(j+1))−(vA(j−1)+vB(j−1))]) (6) Note that the first summand in (5) and (6) is the discounted value of losing the contest at j and the second summand in each of these expressions is the expected gain arising from the contest at j. For at least one player this gain will be zero and for the other player it will be non-negative and strictly positive as long as J(j −1) 6= J(j + 1), where J(l) ≡ vA(l) +vB(l) is the joint present value of being in statel.
Three immediate implications of the above construction are
(i) zA(j)−zB(j) ≥ 0 if and only if J(j −1)−J(j + 1) ≥ 0 with strict inequality in one if and only if in the other.
(ii) zA(j)−zB(j)≥0if and only ifvB(j) =δvB(j−1)andzA(j)−zB(j)≤0 if and only if vA(j) =δvA(j + 1).
(iii) IfzA(j)−zB(j)≥0 thenvA(j) =δ[vA(j−1) +vB(j−1))−vB(j+ 1)], and ifzA(j)−zB(j)≤0thenvB(j) =δ[vA(j+1)+vB(j+1))−vA(j−1)].
By assumption0 and m are terminal states so that vA(0) =ZA ≥ZB = vB(m) andvA(m) = vB(0) = 0. Moreover, since player A can only receive a positive payoffin the state0, playerB can only receive a positive payoffin the state m, and both players have available the opportunity to always expend zero effort, in any Markov perfect equilibrium the following inequalities hold for all j:
0≤vA(j)≤δjZA (7)
0≤vB(j)≤δm−jZB (8) and
vA(j) +vB(j)≤max(δjZA, δm−jZB) (9)
We can now prove the following
Proposition 1 Consider a tug-of-war withm≥3. Supposej0 ∈{2, ...m−1} exists such that
δj0−1ZA > δm−(j0−1)ZB and δj0ZA< δm−j0ZB. (10) Then a unique Markov perfect equilibrium exists which is characterized as follows:
For all interior states j /∈ {j0 − 1, j0}, the equilibrium effort choices are a(j) =b(j) = 0. Only atj0−1and j0 does a battle with a positive probability of stricty positive effort choices take place. Payoffs for A in the continuation game at j areδjZA forj < j0−1, (1−1δ2)[δj0−1ZA−δm−(j0−1)ZB]forj =j0−1, and0forj ≥j0; payoffs forB areδm−jZB forj > j0,(1−1δ2)[δm−j0ZB−δj0ZA] for j =j0 and 0 for j ≤j0−1.
Proof. We consider existence here and relegate the proof of uniqueness to the Appendix. We consider the following candidate equilibrium: For all interior statesj /∈{j0−1, j0}, the effort choices area(j) =b(j) = 0. Atj0−1 and j0 players choose efforts according to cumulative distribution functions Fj andGj for players A and B in states j as follows:
Fj0−1(a) =
⎧⎪
⎨
⎪⎩
a
δ
(1−δ2)∆jBA0 f or a∈[0, δ∆
j0 BA
(1−δ2)] 1 f or a > δ∆
j0 BA
(1−δ2)
(11)
Gj0−1(b) =
⎧⎨
⎩ 1−
δ (1−δ2)∆jBA0
δδj0−2ZA +δδj0−b2ZA f or b∈[0, δ∆
j0 BA
(1−δ2)]
1 f or b > δ∆
j0 BA
(1−δ2)
(12)
Fj0(a) =
⎧⎨
⎩ 1−
δ
(1−δ2) ∆jAB0−1
δδm−(j0+1)ZB + a
δδm−(j0+1)ZB f or a∈[0,δ∆
j0−1 AB
(1−δ2) ]
1 f or a > δ∆
j0−1 AB
(1−δ2)
(13) and
Gj0(b) =
⎧⎪
⎨
⎪⎩
b
δ
(1−δ2)∆jAB0−1 f or b∈[0,δ∆
j0−1 AB
(1−δ2) ] 1 f or b > δ∆
j0−1 AB
(1−δ2) .
(14)
... ...
ZA A j
v =δ 0−2
=0 vB
=?
vA vA=0
( )j ZB
B m v vA
1 0
0+
= −
= δ
=0
vB vB=?
0−2
j j0−1
j0 j0+1
Figure 2:
where
∆jBA0 = [δm−j0ZB−δj0ZA]and ∆jAB0−1 = [δj0−1ZA−δm−(j0−1)ZB]. (15) Notefirst that this equilibrium candidate has the properties described in Proposition 1. Players’ continuation values can be stated as functions of the respective state j as follows:
vA(j) =
⎧⎨
⎩
δjZA for j < j0−1
1
(1−δ2)[δj0−1ZA−δm−(j0−1)ZB] for j =j0−1
0 for j ≥j0
(16) and
vB(j) =
⎧⎨
⎩
δm−jZB for j > j0 1
(1−δ2)[δm−j0ZB−δj0ZA] for j =j0
0 for j ≤j0−1.
(17) These constitute the payoffs stated in the proposition. For 0 < j < j0 −1, playerAwins the nextj battles without any effort. This takesj periods and explains why the value of the final prize must be discounted to δjZA. Also, B does not expend effort in these j battles andfinally loses after j battles.
Hence, B’s payoff is equal to zero. For m > j > j0, players A andB simply switch roles.
Turn now to the states j0−1 andj0 as in Figure 2. We call these states
"tipping states", because of their pivotal role in determining the outcome of the contest. Consider j0 −1. From there, if A wins, the game moves to j0 −2 with continuation values vA(j0 −2) = δj0−2ZA and vB(j0 −2) = 0.
If B wins, the game moves to j0 with continuation values vA(j0) = 0 and vB(j0). Assuming that δj0−2ZA > vB(j0) (which can be confirmed later), and applying the results on the standard all-pay auction, the continuation
values are
vA(j0 −1) = zA(j0−1)−zB(j0−1) =δ[δj0−2ZA−vB(j0)] (18) and vB(j0 −1) = 0, where zA(j0 −1) andzB(j0 −1) denote the prizes that A and B respectively attribute to winning the battle at j0 −1, given the continuation of the game as described in the candidate equilibrium. Similarly, atj0, if Awins, the game moves toj0−1with continuation values vA(j0−1) as in (18) and vB(j0 −1) = 0. If B wins, the game moves to j0 + 1 with continuation valuesvA(j0+ 1) = 0andvB(j0+ 1) =δm−(j0+1)ZB. This yields a continuation value for player B of
vB(j0) =zB(j0)−zA(j0) =δ[δm−(j0+1)ZB−vA(j0−1)], (19) and vA(j0) = 0. The solution to this system of equations yields the posi- tive equilibrium values in the middle lines of (16) and (17), and the zero continuation value in the respective state for the other player.
It remains to be shown that the choices described in the candidate equi- librium indeed describe equilibrium behavior. The one-stage deviation prin- ciple applies here.10 The continuation values (16) and (17) can be used to consider one-stage deviations for A and for B.
A deviation b0(j) > 0 at a state 0 < j < j0 −1 changes the path from moving toj−1in the next period toj+1. However,vB(j−1) =vB(j+1) = 0. Hence, this deviation reduces B’s payoff by b0(j) compared to b(j) = 0. A deviation b0(j)>0 atj > j0 does not change the state in t+ 1 compared to b(j) = 0 in the candidate equilibrium, due to the tiebreaking rule employed.
The deviation reducesB’s payoffbyb0(j)compared tob(j) = 0. An equivalent logic applies for a(j) at statesj /∈{j0 −1, j0}.
Turn now to the state j0. In the candidate equilibrium, in state j0 con- testant A randomizes on the support [0,(1δ
−δ2)[δj0−1ZA−δm−(j0−1)ZB]]. All actions in the equilibrium support forA atj0 yield the same expected payoff equal to Gj0(x)δvA(j0 −1) + (1−Gj0(x))0−x = 0. A possible one-stage deviation forA at j0 is an a0(j0)> (1−δδ2)[δj0−1ZA−δm−(j0−1)ZB]. Compared
10To confirm this it is sufficient to show that the condition of continuity at infinity is fulfilled for this game. We may then apply Theorem 4.2 in Fudenberg and Tirole (1993).
This condition requires that the supremum of the payoffdifference that can emerge from strategies that differ after periodtconverges to zero ast→ ∞. However, a supremum for this isδt[Zi+1−1δK]fori=A, B, and this converges to zero ast→ ∞.
to the action a(j0) = (1−δδ2)[δj0−1ZA−δm−(j0−1)ZB] that is inside A’s equili- brium support, this also leads to statej0−1, but costs the additional amount a0(j0)−a(j0)>0. The deviation is therefore not profitable for A. The same type of argument applies for b(j0).
A similar argument applies to the statej0−1. In the candidate equilibri- um, in statej0−1contestantArandomizes on the support[0,(1δ
−δ2)(δm−j0ZB− δj0ZA)]. All actions in the equilibrium support for A at j0−1 yield the sa- me expected payoff equal to Gj0−1(x)δvA(j0 −2) + (1−Gj0−1(x))0−x =
1
1−δ2[δj0−1ZA−δm−(j0−1)ZB] = vA(j0−1).11 A possible one-stage deviation for A atj0−1 is ana0(j0−1)> (1−δδ2)[δm−j0ZB−δj0ZA]. Compared to the action a(j0 −1) = (1−δδ2)[δm−j0ZB−δj0ZA] that is the upper bound of A’s equilibrium support, this also leads to state j0 −2, but costs the additional amount a0(j0−1)−a(j0−1)>0. The deviation is not profitable forA. The same type of argument applies for b(j0−1).
Intuitively, outside of the statesj0−1andj0, one of the players is indif- ferent between winning and losing the component contest. For instance, in the state j0−2, the best that player B could achieve by winning the next component contest is to enter the statej0−1at whichB’s continuation value is still zero and smaller than player A’s continuation value. As B does not gain anything from reaching j0−1, B should not spend any effort trying to reach this state. But if B does not spend effort to win, it is easy for A to win.
The states j0−1 andj0 are different. Battle victory or defeat at one of these points leads to different continuation games and allocates a considerable rent between A and B. This makes competition particularly strong at these states. We call these states "tipping states"because success of an advantaged player at each of these two states "tips"the game so that victory is obtained without further effort. A loss by the advantaged player throws the system back into a competitive state where the player becomes disadvantaged.
Proposition 1 also shows that the allocation of a prize in a tug-of-war leads to a seemingly peaceful outcome whenever the conflict starts in a state other than a tipping state. This will be important for drawing conclusions in section 3 about the efficiency properties of a tug-of-war as an allocation
11More formally, all actions in the support ofA’s equilibrium local strategy that are not mass points ofB’s local strategy yield the same expected payoff. SinceBhas a mass point at zero, this does not hold ata= 0, but for everyain a neigborhood above zero.
mechanism.
Proposition 1 does not consider all possible parameter cases. Before tur- ning to the remaining cases, note that the casej0 = 1cannot emerge, as this requires δZA< δm−1ZB, and this contradicts ZA ≥ZB for m > 2. However, playerA’s dominance could be sufficiently large that no interiorj0 exists that has the properties defined in Proposition 1. This leads to
Proposition 2 Suppose that δm−1ZA > δZB. Then a unique Markov perfect equilibrium exists withvB(j) = 0 andvA(j) =δjZA for all j ∈{1, ..., m−2}, and vA(m−1) =δm−1ZA−δZB and vB(m−1) = 0 at j =m−1.
Proof.We show that the following effort choices constitute an equilibrium and yield the payoffs described in the proposition. Uniqueness follows the argument in the Appendix.
Effort isa(j) =b(j) = 0for allj ∈Mint\{m−1}and forj =m−1efforts are chosen according to the following cumulative distribution functions:
Fm−1(a) =
½ a
δZB f or a∈[0, δZB] 1 f or a > δZB
(20)
Gm−1(b) =
½ (1− δδZm−B1−ZbA) f or b∈[0, δZB]
1 f or b > δZB. (21)
Note that this behavior yields the payoffs that are characterized in Propositi- on 2. For states j = 1,2, ...,(m−2), Awins after j further battles, and none of the players expends effort. This confirms vA(j) =δjZA andvB(j) = 0 for all j = 1, ...m−2. For j =m, the payoffs are vA(m) = 0 and vB(m) = ZB. Finally, forj =m−1, given the mixed strategies described by (20) and (21), the payoffs are vA(m−1) =δm−1ZA> δZB andvB(m−1) = 0.
Now we confirm that the effort choices in the candidate equilibrium are indeed mutually optimal replies. For interior states j < m−1, a deviation b0(j) > 0 makes B win the battle, instead of A. It leads to j + 1, instead of j −1, but vB(j + 1) = vB(j −1) = 0. Hence, this deviation reduces B’s payoff by b0(j) compared to b(j) = 0. For A, for j < m −1, contestant A reaches j = 0 along the shortest possible series of battle victories and does not spend any effort. Any positive effort can therefore only decrease A’s payoff. For j = m−1, the battle either leads to j = m where B finally wins the prize, or to j =m−2. The values the players attribute to reaching these states are vA(m) = 0, vB(m) = ZB, and vA(m −2) = δm−2ZA and
vB(m−2) = 0. Using the results in Hillman and Riley (1989) and Baye, Kovenock and deVries (1996) on a complete information all-pay auction with prizes δ[δm−2ZA − 0] = δm−1ZA for A and δ[ZB −0] = δZB for B, it is confirmed that (20) and (21) describe the unique equilibrium cumulative distribution functions of effort for this all-pay auction.
Proposition 2 shows that a very strong player has a positive continuation value regardless of the interior state in which the tug-of-war starts and wins with probability 1 without expending effort for every interior state except j =m−1.
So far we have ruled out the case of equality of continuation values at interior states, and we turn to this case now which exhausts the set of possible cases.
Proposition 3 The tug-of-war with δj0ZA = δ(m−j0)ZB ≡ Z for some j0 ∈ {2, ...(m−1)}has a unique subgame perfect equilibrium in which players spend a(j) = b(j) = 0 in all interior states j 6= j0. They choose efforts a(j) and b(j)atj =j0 from the same uniform distribution on the range [0, Z]. Payoffs are vA(j) =δjZA and vB(j) = 0 for j < j0,vA(j) = 0 and vB(j) =δm−jZB
for j > j0 and vA(j) =vB(j) = 0 for j =j0.
Proof.We again construct an equilibrium to demonstrate existence. Un- iqueness follows from arguments similar to those appearing in the Appendix.
In the candidate equilibrium each contestant expends zero effort at any state j 6= j0 and expends effort at j = j0 according to a draw from the distribution
F(x) =
½ x
Z for x∈[0, Z]
1 for x > Z. (22)
At j = j0 the expected effort of each player equals Z/2, and each wins this battle with a probability of 1/2 and, in this case, eventually wins the overall contest j0 −1 or (m−j0)−1 periods later, respectively, without spending any further effort. This determines the continuation values in the candidate equilibrium. These continuation values are
vA=vB = 0 if j =j0
vA=δjZA andvB= 0 if j < j0
vA= 0 and vB =δm−jZB if j > j0.
(23) It remains to show that the candidate equilibrium describes mutually optimal replies. Consider one-stage deviations for A and B for some state
j < j0. A choice a0(j) > 0 will not change the equilibrium outcome in the battle in this period and hence will simply reduce A’s payoff by a0(j). A choice b0(j) > 0 will make B win. If j < j0 −1, following the candidate equilibrium A will simply win a series of battles until final victory occurs.
Hence, b0(j) > 0 reduces B’s payoff by this same amount b0(0) of effort. If j =j0−1, B’s battle victory will lead to j =j0, and candidate equilibrium play from here on will yield a payoff equal to zero to B. Accordingly, the deviation b0(j) > 0 yields a reduction of B’s payoff by this same amount.
Consider one-stage deviations for A and B in some state j > j0. The same line of argument applies, with A and B switching roles. Finally, consider one-stage deviations for A and B at j =j0. Any such deviation for A must be a choice a0(j)> Z. Compared to a(j) =Z, this choice makesA win with the same probability 1, but yields a reduction in A’s payoff by a0(j)−Z, compared to a(j) = Z. The same argument applies for deviations by B at this state.
The intuition for Proposition 3 is as follows. The two contestants enter into a very strong fight whenever they reach the state j = j0. In this state they are perfectly symmetric and they anticipate that the winner of the battle in this state moves straight to final victory. In the battle that takes place in this case, they dissipate the maximum feasible rent from winning this battle.
This maximum rent is what they get if they can move from there through a series of uncontested battles to final victory. Once one of the contestants, sayA, has acquired some advantage in the sense that the contest has moved to j < j0, the only way for B to reach victory passes through the state with j = j0. As all rent is dissipated in the contest that takes place there, B is simply not willing to spend any effort to move the contest to that state.
Hence, the considerable effort that is spent at the point at which the tug-of- war becomes symmetric in terms of the prizes that are at stake for the two contestants prevents the contestant who is lagging behind in terms of battle victories from spending positive effort.
Discounting played two important roles in our analysis. First, discounting leads to payofffunctions that are continuous at infinity, allowing the applica- tion of the one-stage deviation principle, which greatly facilitates our proofs.
Moreover, discounting is essential in giving a meaningful role to the distance to the state with final victory. The following holds:
Proposition 4 For a given value of ZZA
B >1, the tipping state j0 is an incre- asing step function of δ. Moreover, as δ→1, A wins the tug-of-war without
effort starting from any state j < m−1.
Proof.The tipping state j0 is by definition the smallest statej for which playerBis advantaged:j0 = min{j ∈Mint¯¯δjZA≤δm−jZB}when this set is non-empty, and j0 =m otherwise. Forδ > 0, the inequality δjZA≤δm−jZB
is equivalent toδ2j−mZA ≤ZB. Sincem ≥3, forδsufficently close to zero the inequality is clearly satisfied for j =m−1, so that j0 is interior. Moreover, since δ2j−m ≥ 1 for j ≤ m2, it must be the case that j0 > m2. As δ → 1, the inequality is violated at all interior states, even at j = m−1. In this case, by definiton j0 = m, and from Proposition 2 player A wins the war from any state j < m −1. For any 0 < δ < 1, δ2j−mZA ≤ ZB is equivalent to 2j −m ≥ log
ZB ZA
logδ , so that j0 is the smallest index j satisfying the inequality.
Since the left hand side of this inequality is positive, and both the numerator and denominator of the right hand side are negative, as δ increases, the right hand side monotonically increases, eventually diverging to∞asδ →1.
Hence, as δ increases, the smallest index j satisfying the inequality must increase in steps until it hits m.
As the discount factor increases, relative prize value or player strength plays a greater role in the determination of the outcome than distance. For any given value of ZZB
A <1, asδincreases the tipping statej0 moves in discrete jumps towards m. Player A may suffer a greater distance disadvantage and still win the prize with certainty.
3 Expenditure, allocative efficiency and the cost of delay
The tug-of-war withm >2resolves the allocation problem along a sequence of states, where a violent battle may, but need not take place at each state.
Only in the tipping states is positive effort expended with positive probability.
Once the process leaves the tipping states, the war moves to a terminal state, without further effort being expended. A tug-of-war that starts in a tipping state will therefore be called "violent". A tug-of-war that starts outside a tipping state will be called "peaceful".
Compared to the standard all-pay auction, the tug-of-war could be inter- preted as an institution that saves cost of effort in the problem of allocating a prize between rivals who are prepared to expend resources infighting for the
